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Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 746 · 40 questions
TalentJR

How to solve it (Use the identity, skip the guessing): (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²; so a² + b² = (a + b)² − 2ab and (a − b)² = (a + b)² − 4ab. For x² − Sx + P = 0: the sum of the roots is S and the product is P. To factorise x² + Sx + P, find two numbers that multiply to P and add up to S.

NameDateTime takenScore ___ / 40
  1. 1.What are the roots of x² − 10x + 24 = 0? (a) 2 and 12 (b) 1 and 24 (c) 1 and 9 (d) 4 and 6
  2. 2.Using (a + b)² = a² + 2ab + b², what is 83²? (a) 6409 (b) 5929 (c) 6889 (d) 6899
  3. 3.If a + b = 20 and ab = 99, what is (a − b)²? (a) 400 (b) 202 (c) 2 (d) 4
  4. 4.Using (a − b)² = a² − 2ab + b², what is 17²? (a) 529 (b) 299 (c) 391 (d) 289
  5. 5.What are the roots of x² − 13x + 36 = 0? (a) 2 and 18 (b) 1 and 36 (c) 4 and 9 (d) 1 and 12
  6. 6.Which is the correct factorisation of x² + 15x + 56? (a) (x + 7)(x + 8) (b) (x + 1)(x + 56) (c) (x + 1)(x + 14) (d) (x − 7)(x − 8)
  7. 7.Which is the correct factorisation of x² + 9x + 20? (a) (x + 1)(x + 8) (b) (x − 4)(x − 5) (c) (x + 1)(x + 20) (d) (x + 4)(x + 5)
  8. 8.What is the product of the roots of x² − 13x + 42 = 0? (a) 43 (b) 41 (c) 42 (d) 13
  9. 9.What are the roots of x² − 8x + 12 = 0? (a) 1 and 7 (b) 1 and 12 (c) 3 and 4 (d) 2 and 6
  10. 10.What is the sum of the roots of x² − 17x + 72 = 0? (a) 17 (b) 72 (c) 18 (d) 16
  11. 11.If a + b = 17 and ab = 72, what is a² + b²? (a) 289 (b) 217 (c) 433 (d) 145
  12. 12.If a + b = 7 and ab = 10, what is (a − b)²? (a) 49 (b) 29 (c) 9 (d) 3
  13. 13.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 72) (c) (x + 8)(x + 9) (d) (x + 1)(x + 16)
  14. 14.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9999 (b) 9811 (c) 9801 (d) 10201
  15. 15.Using (a − b)² = a² − 2ab + b², what is 46²? (a) 2916 (b) 2126 (c) 2116 (d) 2484
  16. 16.If a + b = 18 and ab = 80, what is a² + b²? (a) 484 (b) 164 (c) 244 (d) 324
  17. 17.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 10) (b) (x − 4)(x − 7) (c) (x + 4)(x + 7) (d) (x + 1)(x + 28)
  18. 18.If a + b = 15 and ab = 56, what is a² + b²? (a) 169 (b) 113 (c) 225 (d) 337
  19. 19.Using (a + b)² = a² + 2ab + b², what is 55²? (a) 2025 (b) 2525 (c) 3035 (d) 3025
  20. 20.What are the roots of x² − 16x + 63 = 0? (a) 7 and 9 (b) 3 and 21 (c) 1 and 63 (d) 1 and 15
  21. 21.If a + b = 14 and ab = 24, what is a² + b²? (a) 196 (b) 148 (c) 172 (d) 244
  22. 22.What are the roots of x² − 15x + 56 = 0? (a) 1 and 56 (b) 2 and 28 (c) 1 and 14 (d) 7 and 8
  23. 23.If a + b = 18 and ab = 77, what is (a − b)²? (a) 324 (b) 4 (c) 170 (d) 16
  24. 24.If a + b = 15 and ab = 50, what is (a − b)²? (a) 225 (b) 125 (c) 5 (d) 25
  25. 25.Which is the correct factorisation of x² + 8x + 15? (a) (x − 3)(x − 5) (b) (x + 1)(x + 15) (c) (x + 1)(x + 7) (d) (x + 3)(x + 5)
  26. 26.What is the sum of the roots of x² − 13x + 40 = 0? (a) 12 (b) 14 (c) 13 (d) 40
  27. 27.If a + b = 6 and ab = 5, what is (a − b)²? (a) 16 (b) 4 (c) 36 (d) 26
  28. 28.If a + b = 15 and ab = 56, what is (a − b)²? (a) 169 (b) 225 (c) 1 (d) 113
  29. 29.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 371 (b) 361 (c) 399 (d) 441
  30. 30.What is the sum of the roots of x² − 9x + 14 = 0? (a) 8 (b) 10 (c) 9 (d) 14
  31. 31.If a + b = 9 and ab = 20, what is a² + b²? (a) 61 (b) 121 (c) 81 (d) 41
  32. 32.What is the sum of the roots of x² − 11x + 18 = 0? (a) 12 (b) 11 (c) 18 (d) 10
  33. 33.If a + b = 8 and ab = 12, what is a² + b²? (a) 88 (b) 64 (c) 52 (d) 40
  34. 34.If a + b = 11 and ab = 24, what is a² + b²? (a) 169 (b) 97 (c) 73 (d) 121
  35. 35.Using (a − b)² = a² − 2ab + b², what is 65²? (a) 4225 (b) 4875 (c) 4235 (d) 5625
  36. 36.Which is the correct factorisation of x² + 16x + 63? (a) (x − 7)(x − 9) (b) (x + 1)(x + 63) (c) (x + 7)(x + 9) (d) (x + 1)(x + 15)
  37. 37.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 7056 (b) 7066 (c) 5776 (d) 6416
  38. 38.What is the sum of the roots of x² − 10x + 24 = 0? (a) 11 (b) 10 (c) 9 (d) 24
  39. 39.What is the sum of the roots of x² − 14x + 45 = 0? (a) 15 (b) 13 (c) 14 (d) 45
  40. 40.If a + b = 11 and ab = 18, what is a² + b²? (a) 157 (b) 103 (c) 121 (d) 85

Answer key

  1. 4 and 6
  2. 6889
  3. 4
  4. 289
  5. 4 and 9
  6. (x + 7)(x + 8)
  7. (x + 4)(x + 5)
  8. 42
  9. 2 and 6
  10. 17
  11. 145
  12. 9
  13. (x + 8)(x + 9)
  14. 9801
  15. 2116
  16. 164
  17. (x + 4)(x + 7)
  18. 113
  19. 3025
  20. 7 and 9
  21. 148
  22. 7 and 8
  23. 16
  24. 25
  25. (x + 3)(x + 5)
  26. 13
  27. 16
  28. 1
  29. 361
  30. 9
  31. 41
  32. 11
  33. 40
  34. 73
  35. 4225
  36. (x + 7)(x + 9)
  37. 7056
  38. 10
  39. 14
  40. 85

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